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Open Access Publications from the University of California

Scholarly Works (2 results)

  • Thesis
  • Peer Reviewed
UCLA Electronic Theses and Dissertations (2015)

Following the techniques initiated in \cite{MP}, we continue to study the limit

shapes of random permutations avoiding a specific subset of patterns.

We consider patterns in $S_3$ extensively, and also prove some results

regarding pairs of permutations with one in $S_3$ and another in $S_4$.

We analyze the limiting distribution of a pattern-avoiding permutation,

and calculate the asymptotic behavior of distributions of positions of numbers

in the permutations. The distributions vary significantly depending on

which patterns are avoided. We also apply our results to obtain results on various

permutation statistics.

    Cover page: Limit Shapes of Restricted Permutations
    • Article
    • Peer Reviewed
    Other (2017)
    For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as well as all patterns of length 3. For monotone patterns we utilize the connection with standard Young tableaux with at most k rows and involutions avoiding a monotone pattern of length k. For every pattern of length 3 we give the bivariate generating function with respect to fixed points for the involutions that avoid that pattern, and where applicable apply tools from analytic combinatorics to extract information about the limiting distribution from the generating function. Many well-known distributions appear.
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